Integrand size = 28, antiderivative size = 524 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b c d^2 \left (8 A c^2 d+b^2 B e-b c (4 B d+9 A e)\right )+\left (16 A c^4 d^3-4 b^4 B e^3+b^3 c e^2 (4 B d+A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+6 A e)\right ) x\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}-\frac {2 \left (16 A c^4 d^3-8 b^4 B e^3+b^3 c e^2 (5 B d+2 A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+4 A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{5/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 d (c d-b e) \left (16 A c^3 d^2+4 b^3 B e^2+b^2 c e (B d-A e)-8 b c^2 d (B d+2 A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{5/2} \sqrt {d+e x} \sqrt {b x+c x^2}} \]
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Time = 0.49 (sec) , antiderivative size = 524, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {832, 857, 729, 113, 111, 118, 117} \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {2 d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (b^2 c e (B d-A e)-8 b c^2 d (2 A e+B d)+16 A c^3 d^2+4 b^3 B e^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{5/2} \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^3 c e^2 (2 A e+5 B d)+b^2 c^2 d e (4 A e+5 B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-8 b^4 B e^3\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{5/2} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 (d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b c d^2 \left (-b c (9 A e+4 B d)+8 A c^2 d+b^2 B e\right )+x \left (b^3 c e^2 (A e+4 B d)+b^2 c^2 d e (6 A e+5 B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-4 b^4 B e^3\right )\right )}{3 b^4 c^2 \sqrt {b x+c x^2}} \]
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Rule 111
Rule 113
Rule 117
Rule 118
Rule 729
Rule 832
Rule 857
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \int \frac {(d+e x)^{3/2} \left (\frac {1}{2} d \left (4 b B c d-8 A c^2 d-b^2 B e+9 A b c e\right )+\frac {1}{2} e \left (2 A c^2 d+4 b^2 B e-b c (B d+A e)\right ) x\right )}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2 c} \\ & = -\frac {2 (d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b c d^2 \left (8 A c^2 d+b^2 B e-b c (4 B d+9 A e)\right )+\left (16 A c^4 d^3-4 b^4 B e^3+b^3 c e^2 (4 B d+A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+6 A e)\right ) x\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}+\frac {4 \int \frac {-\frac {1}{4} b d e \left (8 A c^3 d^2-4 b^3 B e^2+b^2 c e (2 B d+A e)-b c^2 d (4 B d+11 A e)\right )-\frac {1}{4} e \left (16 A c^4 d^3-8 b^4 B e^3+b^3 c e^2 (5 B d+2 A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+4 A e)\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^4 c^2} \\ & = -\frac {2 (d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b c d^2 \left (8 A c^2 d+b^2 B e-b c (4 B d+9 A e)\right )+\left (16 A c^4 d^3-4 b^4 B e^3+b^3 c e^2 (4 B d+A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+6 A e)\right ) x\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}+\frac {\left (d (c d-b e) \left (16 A c^3 d^2+4 b^3 B e^2+b^2 c e (B d-A e)-8 b c^2 d (B d+2 A e)\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^4 c^2}-\frac {\left (16 A c^4 d^3-8 b^4 B e^3+b^3 c e^2 (5 B d+2 A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+4 A e)\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{3 b^4 c^2} \\ & = -\frac {2 (d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b c d^2 \left (8 A c^2 d+b^2 B e-b c (4 B d+9 A e)\right )+\left (16 A c^4 d^3-4 b^4 B e^3+b^3 c e^2 (4 B d+A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+6 A e)\right ) x\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}+\frac {\left (d (c d-b e) \left (16 A c^3 d^2+4 b^3 B e^2+b^2 c e (B d-A e)-8 b c^2 d (B d+2 A e)\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{3 b^4 c^2 \sqrt {b x+c x^2}}-\frac {\left (\left (16 A c^4 d^3-8 b^4 B e^3+b^3 c e^2 (5 B d+2 A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+4 A e)\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{3 b^4 c^2 \sqrt {b x+c x^2}} \\ & = -\frac {2 (d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b c d^2 \left (8 A c^2 d+b^2 B e-b c (4 B d+9 A e)\right )+\left (16 A c^4 d^3-4 b^4 B e^3+b^3 c e^2 (4 B d+A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+6 A e)\right ) x\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}-\frac {\left (\left (16 A c^4 d^3-8 b^4 B e^3+b^3 c e^2 (5 B d+2 A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+4 A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x}\right ) \int \frac {\sqrt {1+\frac {e x}{d}}}{\sqrt {x} \sqrt {1+\frac {c x}{b}}} \, dx}{3 b^4 c^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left (d (c d-b e) \left (16 A c^3 d^2+4 b^3 B e^2+b^2 c e (B d-A e)-8 b c^2 d (B d+2 A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}} \, dx}{3 b^4 c^2 \sqrt {d+e x} \sqrt {b x+c x^2}} \\ & = -\frac {2 (d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b c d^2 \left (8 A c^2 d+b^2 B e-b c (4 B d+9 A e)\right )+\left (16 A c^4 d^3-4 b^4 B e^3+b^3 c e^2 (4 B d+A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+6 A e)\right ) x\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}-\frac {2 \left (16 A c^4 d^3-8 b^4 B e^3+b^3 c e^2 (5 B d+2 A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+4 A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{5/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 d (c d-b e) \left (16 A c^3 d^2+4 b^3 B e^2+b^2 c e (B d-A e)-8 b c^2 d (B d+2 A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{5/2} \sqrt {d+e x} \sqrt {b x+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 25.17 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.01 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \left (b (d+e x) \left (b (b B-A c) (c d-b e)^3 x^2+(c d-b e)^2 \left (-8 A c^2 d+5 b^2 B e+b c (5 B d-2 A e)\right ) x^2 (b+c x)+A b c^2 d^3 (b+c x)^2+c^2 d^2 (3 b B d-8 A c d+10 A b e) x (b+c x)^2\right )+\sqrt {\frac {b}{c}} x (b+c x) \left (\sqrt {\frac {b}{c}} \left (16 A c^4 d^3-8 b^4 B e^3+b^3 c e^2 (5 B d+2 A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+4 A e)\right ) (b+c x) (d+e x)+i b e \left (16 A c^4 d^3-8 b^4 B e^3+b^3 c e^2 (5 B d+2 A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+4 A e)\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e (c d-b e) \left (8 A c^3 d^2+8 b^3 B e^2-b^2 c e (B d+2 A e)-b c^2 d (4 B d+5 A e)\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{3 b^5 c^2 (x (b+c x))^{3/2} \sqrt {d+e x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1132\) vs. \(2(470)=940\).
Time = 2.28 (sec) , antiderivative size = 1133, normalized size of antiderivative = 2.16
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1133\) |
default | \(\text {Expression too large to display}\) | \(3247\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.18 (sec) , antiderivative size = 1266, normalized size of antiderivative = 2.42 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {7}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {7}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{7/2}}{{\left (c\,x^2+b\,x\right )}^{5/2}} \,d x \]
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